Measures of dispersion Absolute measure, relative measures Range of Coe. of Range Mean deviation and coe. of mean deviation Quartile deviation IQR, coefficient of QD Standard deviation and coefficient of variation

1. Jagdish Powar
Statistician cum tutor
Communicate Medicine
SMBT, IMSRC, Nashik.
JDP-CM-SMBT 1

2. Competency & Learning objectives
Competency Learning Objectives
CM6.4
Enumerate, discuss and demonstrate
Common sampling techniques, simple
statistical methods, frequency
distribution, measures of central
tendency and dispersion
The student should be able to
 Define measures of dispersion
 Calculate different measures of
dispersion (SD, Coefficient of
variation, Mean Deviation)
 Comment on variation of data
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3. What is measure of dispersion?
• Central tendency measures do not reveal the
variability present in the data. Dispersion is measure
of variation
• Dispersion is the scattered ness of the data series
around it average.
• Dispersion is the extent to which values in a
distribution differ from the average of the
distribution.
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4. Requirement of good measures of dispersion.
1. It should be rigidly defined.
2. It should be easy to understand and easy to calculate.
3. It should be based on all the observations of the data.
4. It should be least affected by the sampling
fluctuation.
5. It should not be unduly affected by the extreme
values.
6. It should be capable of further mathematical
treatment and statistical analysis
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5. Absolute Measure and Relative Measure
Absolute Measure-
Measure of the dispersion in the original unit of the
data. Variability distribution can be compared
provided they are given in the same unit and have
the same average
Relative Measure-
It is the ratio of absolute measures and unit free
measurement. This ratio is known as coefficient of
absolute dispersion.
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6. Measures of Dispersion
Following are the different measures of
dispersion
1. Range & Coefficient of range
2. Mean Deviation & Coefficient of Mean Deviation
3. Quartile deviation & Coefficient of Quartile
deviation
4. Standard Deviation & Coefficient of Variation
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7. Range
It is the simplest possible measure of dispersion and is
defined as the difference between the largest and
smallest values of the variable and it is given as
R= L-S
L= Largest observation, S= Smallest observation
Coefficient of Range
Coefficient of range is given as
Coe. of R=
𝑳−𝑺
𝑳+𝑺
Range & Coefficient of Range
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8. Mean Deviation(MD) & Coefficient of MD
Mean deviation
It is the average of the absolute values of the deviation from
the mean. Mean deviation taken from mean is given as
For ungrouped data:-
Mean deviation (MD) =
𝛴𝐼𝑥−x̅ 𝐼
𝑛
For grouped data:-
Mean deviation (MD) =
𝛴𝑓∗𝐼𝑥−x̅ 𝐼
𝛴𝑓
where x̅= mean
f=frequency
Coefficient of Mean Deviation
Coe. of MD=
𝑀𝑒𝑎𝑛 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑀𝑒𝑎𝑛
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9. Quartiles & Quartile deviation
Quartile divides the data into four equal parts, there are three such
observation divides the data into four equal part denoted as Q1, Q2, Q3
Q1 = First Quartile =
𝑛+1
4
𝑡ℎ 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛
Q2 = Second Quartile = 2∗
𝑛+1
4
𝑡ℎ 𝑜𝑟𝑑𝑒𝑟𝑒𝑑𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 =Median
Q3 = Third Quartile = 3*
𝑛+1
4
𝑡ℎ 𝑜𝑟𝑑𝑒𝑟𝑒𝑑𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛
IQR, QD & Coefficient of QD
• Inter quartile range (IQR) =Q3 – Q1
• Quartile deviation (QD) =
Q3 –
Q1
2
• Coefficient of QD =
Q3 –
Q1
Q3 +
Q1
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10. Deciles & Percentiles
Deciles:-
Deciles are the observations which divides the data into 10
equal parts. There are 9 such observation divides data into 10
parts, this observations are denoted D1, D2,……….., D8, D9
Percentiles:-
Percentiles are the observations which divides the data into
100 equal parts. There are 99 such observation divides data
into 10 parts, this observations are denoted P1, P2,……….., P98,
P99
Median= Q2 = D5 = P50
Q1=P25, Q3=P75
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11. Standard Deviation & Coe. Of variation
Standard deviation is the square root of the arithmetic mean of the
squares of deviation of its items from their arithmetic mean.
Ungrouped data:-
SD =
Σ 𝑥− 𝑥 2
𝑛
when n > 30
=
Σ 𝑥− 𝑥 2
𝑛−1
when n < 30
Grouped data
SD =
𝛴𝑓∗ 𝑥−x̅ 2
𝛴𝑓
Variance = SD2
Coe. of Variation(CV)=
𝑆𝐷
𝑀𝑒𝑎𝑛
∗ 100
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12. Exercises…….
1) Following are the diastolic blood pressure (mmHg) of 8
individuals. Calculate Range , Coe. of range, Mean deviation,
Coe. of mean deviation, SD and Coe. of variation
90, 82, 80, 92, 80, 72, 78, 82
Ans-
Let us denote X-diastolic blood pressure
R= L-S
L= largest observation=92, S= Smallest observation=72
R= 92-72=20
Coe. of R=
𝐿−𝑆
𝐿+𝑆
=
92−72
92+72
=
20
164
=0.12
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13. Mean deviation & Coe. of MD:-
x̅ =
𝛴𝑥
𝑛
=
656
8
=82
Mean deviation (MD) =
𝛴𝐼𝑥−x̅ 𝐼
𝑛
=
36
8
=4.5
Coe. of MD=
𝑀𝑒𝑎𝑛 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑀𝑒𝑎𝑛
=
4.5
82
=0.0549
X 𝑥−x̅ 𝐼𝑥−x̅ 𝐼
90 8 8
82 0 0
80 -2 2
92 10 10
80 -2 2
72 -10 10
78 -4 4
82 0 0
𝛴𝑥=656 𝛴𝐼𝑥−x̅ 𝐼 =36
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14. SD, Variance &Coe. of variation
x̅ =
𝛴𝑥
𝑛
=
656
8
=82
SD =
Σ 𝑥− 𝑥 2
𝑛−1
=
288
7
= 41.14 =6.41
Variance = SD2 =41.14
Coe. of Variation(CV)=
𝑆𝐷
𝑀𝑒𝑎𝑛
∗ 100
=
6.41
82
∗ 100=7.81%
X 𝑥−x̅ 𝑥 − 𝑥 2
90 8 64
82 0 0
80 -2 4
92 10 100
80 -2 4
72 -10 100
78 -4 16
82 0 0
𝛴𝑥=656 𝛴 𝑥 − 𝑥 2
=288
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15. 2) Following table gives the age distribution of patients from RHTC, SMBT-
Ekdara
Calculate SD, Variance and Coe. of Variation.
Ans-
Mean ( 𝑥) =
𝛴𝑓∗𝑥
𝑁
=
6400
160
=40
SD =
𝛴𝑓∗ 𝑥−x̅ 2
𝛴𝑓
=
29000
160
= 181.25 =13.46
Variance = SD2 =181.25
Coe. of Variation(CV)=
𝑆𝐷
𝑀𝑒𝑎𝑛
∗ 100
=
13.46
40
∗ 100 = 33.65%
Age (Years) 10-20 20-30 30-40 40-50 50-60 60-70
No. of patients 15 25 30 55 25 10
15
JDP-CM-SMBT
CI f x f*x 𝒙 − 𝒙 𝟐 f* 𝒙 − 𝒙 𝟐
10 - 20 15 15 225 625 9375
20 - 30 25 25 625 225 5625
30 - 40 30 35 1050 25 750
40 - 50 55 45 2475 25 1375
50 - 60 25 55 1375 225 5625
60 - 70 10 65 650 625 6250
∑ 160 6400 29000

16. 3) The mean systolic blood pressure and hemoglobin levels
of a group of 100 individuals is given below. Comment on
coefficient of variation.
Ans-
Coe. of Variation(CV)=
𝑆𝐷
𝑀𝑒𝑎𝑛
∗ 100
CV(SBP) =
𝑆𝐷
𝑀𝑒𝑎𝑛
∗ 100=
9.8
132
∗ 100=7.42%
CV(Hb) =
𝑆𝐷
𝑀𝑒𝑎𝑛
∗ 100=
1.3
12.6
∗ 100=10.32%
CV(Hb) > CV(SBP)
Systolic blood pressure shows more variatio
Variable Mean SD
Systolic blood pressure (mmHg) 132 9.8
Blood hemoglobin (gm%) 12.6 1.3
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17. JDP-CM-SMBT 17